Electricity is made up of two components, the voltage, and the current.
A device requires a specific voltage to operate at - any higher than this and the device could blow up, any lower and the device won't work right.
Your servos require 5V to operate. Any more, and the coils could burn out. Any less, and the motors won't be able to turn well enough.
Then there is the current. This, coupled with the voltage, defines the amount of "power" a device consumes. A device has a current requirement, which is how much current it wants to pull from the power source. A power source can only supply up to a limited amount of current, so if your device tries to pull more than the available current from the power source it could break the power source. A device can happily pull less than the power source's current limit.
This is where an Arduino IO pin and a battery differ vastly. A battery has a much higher current limit than an Arduino IO pin. Some batteries (such as car batteries) are capable of delivering many hundreds of amps of current. Small batteries like AA, AAA, C, D, PP3, etc can supply much lower currents, but are still many more times more powerful than an Arduino IO pin.
An Arduino is capable of supplying no more than 40mA of current through any one IO pin, and the total current drawn by the ATMega328P chip (including what is drawn from its IO pins) must not exceed 200mA. LEDs typically draw around 20 to 25mA, so they are fine to connect to an IO pin for their power (with a suitable resistor to limit their current - they have no internal resistance of their own), but anything more powerful, such as a relay, or a motor, must be powered by some other power source. This could be the Arduino's 5V pin, if you don't need more than a few hundred mA, or an external battery or power supply.
Servos are slightly different to other motors in that they have a separate control line to their power lines, so you can connect the power up to an external source and control them through an Arduino IO pin.
So yes, using an external power source to provide the power to the servos, you could theoretically control as many servos as there are IO pins. You could theoretically control many many more by adding IO expanders and other controller chips.